English
For any prime p and integer x, v_p^{ℤ}(x) = 1 if and only if p ∤ x.
Русский
Для любого p и целого x, v_p^{ℤ}(x) = 1 тогда и только тогда, когда p не делит x.
LaTeX
$$v_p^{ℤ}(x) = 1 \\Leftrightarrow p \\nmid x$$
Lean4
theorem padicValuation_eq_one_iff {p : ℕ} [Fact p.Prime] {x : ℤ} : Int.padicValuation p x = 1 ↔ ¬(p : ℤ) ∣ x :=
by
simp only [← Rat.padicValuation_cast, Rat.padicValuation, Valuation.coe_mk, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
Rat.intCast_eq_zero, padicValRat.of_int]
split_ifs
· simp_all
· rw [← exp_zero, exp_injective.eq_iff]
simp_all [Nat.Prime.ne_one Fact.out]